Massey higher products

In this paper, we shall investigate some properties of a class of higher order cohomology operations of several variables.These operations, the higher products, were defined by Massey as a generalization of his triple product.There is a correspondence between the higher products and iterated Whitehead products in homotopy groups [5], [11].It has been noted that the differentials in certain spectral sequences involving the Ext and Tor functors are related to Massey higher products [6], [9].In particular, the differentials in a spectral sequence relating the cohomology of a space with that of its space of loops are generalized higher products.In the first part we establish a number of properties of these higher products.These properties indicate the similarities between the higher products and the cup product.In the final section, the higher products are specialized to an operation of one variable.In certain cases, this operation may be evaluated in terms of primary Steenrod operations (Theorems 14, 19).These results are useful in the computation of the higher product structure of a space with coefficients in a field.1. Definitions.Throughout this paper let (X,A) be a pair of topological spaces and let R be a commutative ring with identity.Also let uu-,uk be positive dimensional cohomology classes, of dimensions Py,---,pk respectively, in the singular cohomology ring H*(X,A;R).Finally, let p(i,j)= 2ZJr = i(pr-1).Under certain conditions, we will be able to define the Massey fc-fold product <[uy,---,uky as a subset of Hpll'k) + 2(X,A;R).We shall first describe a "higher operation" from a subset of the singular cochains to a subset of a cohomology group.Let C*(X, A ; R) be the singular cochain complex with the usual associative cup product pairing.Let ax,---,ak be cocycle representatives of ut, ■•■,uk respectively.If a e CP(X, A ; R), then the symbol à will denote ( -l)"a.Definition 1.A collection of cochains, A = (a(i,j)), for 1 ^ i g j ^ fc and